If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10} b$, what is the median of the set $\{0, 1, a, b,
1/b\}$?
Answer: Because $b<10^b$ for all $b>0$, it follows that $\log_{10}b<b$.  If $b\geq 1$, then $0<\left(\log_{10}b\right)/b^2<1$, so $a$ cannot be an integer. Therefore $0<b<1$, so $\log_{10}b<0$ and $a =
\left(\log_{10}b\right)/b^2<0$.  Thus $a<0<b<1<1/b$, and the median of the set is $\boxed{b}$.

Note that the conditions of the problem can be met with $b = 0.1$ and $a = -100$.